A round-robin tournament of n contestants is one in which each of the (2) pairs of contestants plays each other exactly once, with the outcome of any play being that one of the contestants wins and the other loses Suppose the players are initially numbered 1, 2,.,n. The permutation ,, i, is called a Hamiltonian permutation if i beats i, iz beats is, , and i-, beats i,. Show that there is an outcome of the round-robin for which the number of Hamiltonians is at least n!/2"-1. (Hint. Use the probabilistic method.)